Polynomial Division Calculator
Enter the coefficients of your dividend and divisor polynomials, choose the degree of each, and get the quotient, remainder, and a complete step-by-step long division worked out with your actual numbers. The remainder theorem check tells you instantly whether the divisor is a factor.
Formula
Worked example
Divide P(x) = x^3 - 2x^2 - 4x + 4 by Q(x) = x - 2: step 1, x^3 / x = x^2; step 2, x^2 * (x - 2) = x^3 - 2x^2, subtract to get -4x + 4; step 3, -4x / x = -4; step 4, -4 * (x - 2) = -4x + 8, subtract to get -4. Quotient = x^2 - 4, Remainder = -4.
What is polynomial long division?
Polynomial long division is the process of dividing one polynomial (the dividend, P(x)) by another (the divisor, Q(x)) to produce a quotient A(x) and a remainder R(x), satisfying the equation P(x) = A(x) * Q(x) + R(x). The degree of the remainder must be strictly less than the degree of the divisor, just as in ordinary integer division the remainder is smaller than the divisor. The procedure mirrors the familiar algorithm for dividing integers: at each step you divide the leading term of the current dividend by the leading term of the divisor, multiply that result back through the divisor, and subtract it from the current dividend, repeating until the remainder has degree less than the divisor. The step-by-step panel in this calculator shows every sub-step with your actual coefficients.
The remainder theorem and factor theorem
When you divide P(x) by a linear divisor (x - r), the remainder theorem says the remainder is exactly P(r) - the value of P(x) evaluated at x = r. This is useful for checking: substitute r into P(x) and the result should match the remainder shown by the calculator. The factor theorem is the special case where P(r) = 0, which means the remainder is zero and (x - r) is an exact factor of P(x). You can use this to find factors by testing candidate roots (usually the rational roots given by the rational root theorem) and confirming with polynomial division. The "Is divisor a factor?" output above tells you instantly whether the remainder is zero for any divisor, not just linear ones.
Synthetic division for linear divisors
When the divisor is exactly linear, that is, of the form (ax + b), there is a shortcut called synthetic division that avoids writing out the full polynomial expressions. For a monic divisor (x - r), you write only the coefficients of P(x), carry the leading coefficient down, then repeatedly multiply by r and add to the next coefficient. The bottom row gives the coefficients of the quotient, and the final number is the remainder. This calculator uses full long division internally for generality, but the steps panel notes the synthetic division interpretation for linear divisors. Synthetic division is faster by hand and is a standard technique on standardized algebra tests.
Practical applications of polynomial division
Polynomial division appears throughout algebra, calculus, and engineering. In algebra, it is the first step in partial fraction decomposition, where you split a rational function into simpler pieces for integration. In calculus, you divide before integrating rational functions to remove terms with degree in the numerator greater than or equal to the degree in the denominator. In control engineering, polynomial division is used to analyze transfer functions. Polynomial division also underlies the Euclidean algorithm for polynomials, which finds the greatest common divisor of two polynomials - a key step in simplifying rational expressions and solving systems of polynomial equations.
Polynomial degree names and long-division outcomes
| Degree | Name | Example form | Quotient degree when dividing by linear |
|---|---|---|---|
| 0 | Constant | a | N/A (Q degree must be <= P degree) |
| 1 | Linear | ax + b | 0 (constant quotient) |
| 2 | Quadratic | ax^2 + bx + c | 1 (linear quotient) |
| 3 | Cubic | ax^3 + bx^2 + cx + d | 2 (quadratic quotient) |
| 4 | Quartic | ax^4 + ... + e | 3 (cubic quotient) |
| 5 | Quintic | ax^5 + ... + f | 4 (quartic quotient) |
The degree of the quotient is always deg(P) minus deg(Q) when deg(Q) is less than or equal to deg(P).
Frequently asked questions
How does polynomial long division work?
You divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply that term by the entire divisor, subtract the result from the dividend, and repeat with the new smaller polynomial. Continue until the degree of the remainder falls below the degree of the divisor. This calculator shows every sub-step so you can follow along.
What does the remainder tell me?
When you divide P(x) by Q(x), the remainder R(x) is the part that does not divide evenly. If Q(x) is linear, x - r, the remainder is P(r) by the remainder theorem. A remainder of zero means Q(x) is an exact factor of P(x). When the remainder is non-zero, the full result is P(x) = A(x) * Q(x) + R(x), where A(x) is the quotient.
What is the factor theorem?
The factor theorem states that a linear polynomial (x - r) is a factor of P(x) if and only if P(r) = 0. Equivalently, when you divide P(x) by (x - r) and the remainder is zero, (x - r) is a factor. This calculator checks this automatically and reports "Yes" or "No" in the factor check output.
When can I use synthetic division instead of long division?
Synthetic division works only when the divisor is a linear polynomial (degree 1). It is a shorthand version of long division that uses only the coefficients, making it faster to compute by hand. This calculator performs full long division in every case and notes the synthetic division interpretation when the divisor is linear.
What if the divisor has a higher degree than the dividend?
If the degree of the divisor is greater than the degree of the dividend, the quotient is zero and the entire dividend is the remainder. Division is technically possible but trivial: P(x) = 0 * Q(x) + P(x). The calculator handles this case and reports it clearly.
How do I enter a polynomial like 2x^3 - x + 5?
Set the dividend degree to 3, then enter 2 for the coefficient of x^3, 0 for the coefficient of x^2 (since that term is absent), -1 for the coefficient of x, and 5 for the constant term. The calculator always requires all coefficients down to the constant term, using 0 for any missing powers.